Journal of Nonlinear Science

, Volume 24, Issue 5, pp 769–808 | Cite as

Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

Article

Abstract

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space \(\mathrm{Diff }_{1}(\mathbb R)\) equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat \(L^2\)-metric. Here \(\mathrm{Diff }_{1}(\mathbb R)\) denotes the extension of the group of all compactly supported, rapidly decreasing, or \(W^{\infty ,1}\)-diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter–Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa–Holm equation.

Keywords

Diffeomorphism group Geodesic equation Sobolev H1-metric R-map 

Mathematics Subject Classification

Primary 35Q31 58B20 58D05 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Martin Bauer
    • 1
  • Martins Bruveris
    • 2
  • Peter W. Michor
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Institut de mathématiquesEPFL LausanneSwitzerland

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